\(\int \frac {(1+c^2 x^2)^{3/2}}{a+b \text {arcsinh}(c x)} \, dx\) [360]

Optimal result

Integrand size = 24, antiderivative size = 144 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b c}+\frac {3 \log (a+b \text {arcsinh}(c x))}{8 b c}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b c} \] Output:

1/2*cosh(2*a/b)*Chi(2*(a+b*arcsinh(c*x))/b)/b/c+1/8*cosh(4*a/b)*Chi(4*(a+b 
*arcsinh(c*x))/b)/b/c+3/8*ln(a+b*arcsinh(c*x))/b/c-1/2*sinh(2*a/b)*Shi(2*( 
a+b*arcsinh(c*x))/b)/b/c-1/8*sinh(4*a/b)*Shi(4*(a+b*arcsinh(c*x))/b)/b/c
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+3 \log (a+b \text {arcsinh}(c x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{8 b c} \] Input:

Integrate[(1 + c^2*x^2)^(3/2)/(a + b*ArcSinh[c*x]),x]
 

Output:

(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + Cosh[(4*a)/b]*Cosh 
Integral[4*(a/b + ArcSinh[c*x])] + 3*Log[a + b*ArcSinh[c*x]] - 4*Sinh[(2*a 
)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - Sinh[(4*a)/b]*SinhIntegral[4*( 
a/b + ArcSinh[c*x])])/(8*b*c)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6206, 3042, 3793, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (c^2 x^2+1\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx\)

\(\Big \downarrow \) 6206

\(\displaystyle \frac {\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )^4}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 (a+b \text {arcsinh}(c x))}+\frac {3}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {3}{8} \log (a+b \text {arcsinh}(c x))}{b c}\)

Input:

Int[(1 + c^2*x^2)^(3/2)/(a + b*ArcSinh[c*x]),x]
 

Output:

((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/2 + (Cosh[(4*a)/ 
b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8 + (3*Log[a + b*ArcSinh[c*x] 
])/8 - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/2 - (Sinh[ 
(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/8)/(b*c)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6206

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Subst[Int 
[x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
 

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {{\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arcsinh}\left (x c \right )+\frac {4 a}{b}\right )+4 \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right )+4 \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right )+{\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arcsinh}\left (x c \right )-\frac {4 a}{b}\right )-6 \ln \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c b}\)	\(115\)

Input:

int((c^2*x^2+1)^(3/2)/(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

-1/16*(exp(4*a/b)*Ei(1,4*arcsinh(x*c)+4*a/b)+4*exp(2*a/b)*Ei(1,2*arcsinh(x 
*c)+2*a/b)+4*exp(-2*a/b)*Ei(1,-2*arcsinh(x*c)-2*a/b)+exp(-4*a/b)*Ei(1,-4*a 
rcsinh(x*c)-4*a/b)-6*ln(a+b*arcsinh(x*c)))/c/b
 

Fricas [F]

\[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \] Input:

integrate((c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")
 

Output:

integral((c^2*x^2 + 1)^(3/2)/(b*arcsinh(c*x) + a), x)
 

Sympy [F]

\[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \] Input:

integrate((c**2*x**2+1)**(3/2)/(a+b*asinh(c*x)),x)
 

Output:

Integral((c**2*x**2 + 1)**(3/2)/(a + b*asinh(c*x)), x)
 

Maxima [F]

\[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \] Input:

integrate((c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")
 

Output:

integrate((c^2*x^2 + 1)^(3/2)/(b*arcsinh(c*x) + a), x)
 

Giac [F]

\[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \] Input:

integrate((c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="giac")
 

Output:

integrate((c^2*x^2 + 1)^(3/2)/(b*arcsinh(c*x) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {{\left (c^2\,x^2+1\right )}^{3/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \] Input:

int((c^2*x^2 + 1)^(3/2)/(a + b*asinh(c*x)),x)
 

Output:

int((c^2*x^2 + 1)^(3/2)/(a + b*asinh(c*x)), x)
 

Reduce [F]

\[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\sqrt {c^{2} x^{2}+1}}{\mathit {asinh} \left (c x \right ) b +a}d x +\left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{2}}{\mathit {asinh} \left (c x \right ) b +a}d x \right ) c^{2} \] Input:

int((c^2*x^2+1)^(3/2)/(a+b*asinh(c*x)),x)
 

Output:

int(sqrt(c**2*x**2 + 1)/(asinh(c*x)*b + a),x) + int((sqrt(c**2*x**2 + 1)*x 
**2)/(asinh(c*x)*b + a),x)*c**2